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An integer is the number zero (), a positive
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
(, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The integers form the smallest group and the smallest ring containing the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.


History

The word integer comes from the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''integer'' meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). " Entire" derives from the same origin via the French word '' entier'', which means both ''entire'' and ''integer''. Historically the term was used for a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
that was a multiple of 1, or to the whole part of a mixed number. Only positive integers were considered, making the term synonymous with the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. The definition of integer expanded over time to include
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s as their usefulness was recognized. For example
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in his 1765 '' Elements of Algebra'' defined integers to include both positive and negative numbers. However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century. The use of the letter Z to denote the set of integers comes from the German word '' Zahlen'' ("number") and has been attributed to David Hilbert. The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947. The notation was not adopted immediately, for example another textbook used the letter J and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers. The symbol \mathbb is often annotated to denote various sets, with varying usage amongst different authors: \mathbb^+,\mathbb_+ or \mathbb^ for the positive integers, \mathbb^ or \mathbb^ for non-negative integers, and \mathbb^ for non-zero integers. Some authors use \mathbb^ for non-zero integers, while others use it for non-negative integers, or for (the group of units of \mathbb). Additionally, \mathbb_ is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008 The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that "whole numbers" referred to the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, excluding negative numbers, while "integer" included the negative numbers. "Whole number" remains ambiguous to the present day.


Algebraic properties

Like the natural numbers, \mathbb is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), \mathbb, unlike the natural numbers, is also closed under subtraction. The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring \mathbb. \mathbb is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under
exponentiation Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...
, the integers are not (since the result can be a fraction when the exponent is negative). The following table lists some of the basic properties of addition and multiplication for any integers , and : The first five properties listed above for addition say that \mathbb, under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum or . In fact, \mathbb under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to \mathbb. The first four properties listed above for multiplication say that \mathbb under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that \mathbb under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that \mathbb together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in \mathbb for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
in certain rings. The lack of zero divisors in the integers (last property in the table) means that the commutative ring \mathbb is an integral domain. The lack of multiplicative inverses, which is equivalent to the fact that \mathbb is not closed under division, means that \mathbb is ''not'' a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes \mathbb as its subring. Although ordinary division is not defined on \mathbb, the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
of . The integer is called the ''quotient'' and is called the '' remainder'' of the division of by . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. The above says that \mathbb is a Euclidean domain. This implies that \mathbb is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way. This is the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
.


Order-theoretic properties

\mathbb is a totally ordered set without upper or lower bound. The ordering of \mathbb is given by: An integer is ''positive'' if it is greater than
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
, and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if and , then # if and , then . Thus it follows that \mathbb together with the above ordering is an ordered ring. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.


Construction


Traditional development

In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers,
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
, and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms, call this P. Then construct a set P^- which is disjoint from P and in one-to-one correspondence with P via a function \psi. For example, take P^- to be the ordered pairs (1,n) with the mapping \psi = n \mapsto (1,n). Finally let 0 be some object not in P or P^-, for example the ordered pair (0,0). Then the integers are defined to be the union P \cup P^- \cup \. The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows: -x = \begin \psi(x), & \text x \in P \\ \psi^(x), & \text x \in P^- \\ 0, & \text x = 0 \end The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.


Equivalence classes of ordered pairs

In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s . The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation on these pairs with the following rule: :(a,b) \sim (c,d) precisely when :a + d = b + c. Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has: : a,b)+ c,d):= a+c,b+d) : a,b)cdot c,d):= ac+bd,ad+bc) The negation (or additive inverse) of an integer is obtained by reversing the order of the pair: :- a,b):= b,a) Hence subtraction can be defined as the addition of the additive inverse: : a,b)- c,d):= a+d,b+c) The standard ordering on the integers is given by: : a,b)< c,d)/math> if and only if a+d < b+c. It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form or (or both at once). The natural number is identified with the class (i.e., the natural numbers are embedded into the integers by map sending to ), and the class is denoted (this covers all remaining classes, and gives the class a second time since Thus, is denoted by :\begin a - b, & \mbox a \ge b \\ -(b - a), & \mbox a < b. \end If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as . Some examples are: :\begin 0 &= 0,0)&= 1,1)&= \cdots & &= k,k)\\ 1 &= 1,0)&= 2,1)&= \cdots & &= k+1,k)\\ -1 &= 0,1)&= 1,2)&= \cdots & &= k,k+1)\\ 2 &= 2,0)&= 3,1)&= \cdots & &= k+2,k)\\ -2 &= 0,2)&= 1,3)&= \cdots & &= k,k+2) \end


Other approaches

In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines. Integers are represented as algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, which are assumed to be already constructed (using, say, the Peano approach). There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms. The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair(x,y) that takes as arguments two natural numbers x and y, and returns an integer (equal to x-y). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the proof assistant Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.


Computer science

An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted ''int'' or Integer in several programming languages (such as Algol68, C,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
,
Delphi Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracle ...
, etc.). Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).


Cardinality

The cardinality of the set of integers is equal to ( aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from \mathbb to \mathbb= \. Such a function may be defined as :f(x) = \begin -2x, & \mbox x \leq 0\\ 2x-1, & \mbox x > 0, \end with graph (set of the pairs (x, f(x)) is :. Its inverse function is defined by :\beging(2x) = -x\\g(2x-1)=x, \end with graph :.


See also

* Canonical factorization of a positive integer * Hyperinteger * Integer complexity * Integer lattice * Integer part * Integer sequence * Integer-valued function * Mathematical symbols * Parity (mathematics) * Profinite integer


Footnotes


References


Sources

* ) * * *


External links

*
The Positive Integers – divisor tables and numeral representation tools

On-Line Encyclopedia of Integer Sequences
cf OEIS * {{Authority control Elementary mathematics Abelian group theory Ring theory Elementary number theory Algebraic number theory Sets of real numbers